Online Forecasting of Total-Variation-bounded Sequences

We consider the problem of online forecasting of sequences of length n with total-variation at most C_n using observations contaminated by independent \sigma -subgaussian noise. We also prove a lower bound that matches the upper bound in all parameters (up to a \log(n) factor). To the best of our knowledge, this is the first **polynomial-time** algorithm that achieves the optimal O(n {1/3}) rate in forecasting total variation bounded sequences and the first algorithm that **adapts to unknown** C_n .Our proof techniques leverage the special localized structure of Haar wavelet basis and the adaptivity to unknown smoothness parameters in the classical wavelet smoothing [Donoho et al., 1998]. We also compare our model to the rich literature of dynamic regret minimization and nonstationary stochastic optimization, where our problem can be treated as a special case. We show that the workhorse in those settings --- online gradient descent and its variants with a fixed restarting schedule --- are instances of a class of **linear forecasters** that require a suboptimal regret of \tilde{\Omega}(\sqrt{n}) .